Defining \(\mathcal {K}(f_i,f_j)\) : The soft set intersection cardinality \(\mu (A \cap B)\)  may be adapted to a specific task by defining \(\mathcal {K}(f_{i},f_{j})\)  according to the representation of elements \(f_{i}\) . In the context of this article, a set element \(f_{i}\)  is a 3D scale-invariant keypoint \(f_{i} = \lbrace  \bar{a_{i}}, \bar{g_{i}} \rbrace \)  as described in [1]}, where \(\bar{g_{i}}\)  and \(\bar{a_{i}}\)  are descriptors of local keypoint geometry and appearance, respectively. Keypoint geometry \(\bar{g_{i}} = \lbrace  \bar{x_{i}},\sigma _{i} \rbrace \)  consists of 3D location \(\bar{x_{i}}\)  and scale \(\sigma _{i}\) , and appearance \(\bar{a_{i}}\)  is a vector of local image information, i.e. a rank-ordered histogram of oriented gradients (HOG) [2]}.
